Transcript x - Elsevier

1
A Primer on
Financial Time Series Analysis
Elements of
Financial Risk Management
Chapter 3
Peter Christoffersen
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
2
Overview
Topics under discussion in this Chapter
•
•
•
•
Probability Distributions and Moments
The Linear Model
Univariate Time Series Models
Multivariate Time Series Models
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Common pitfalls encountered while
dealing with time series data
• Spurious detection of mean-reversion
• Spurious regression
• Spurious detection of causality
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
3
4
Univariate Probability Distributions
• Let
denote the cumulative probability
distribution of the random variable .
• The probability of being less than is given by
• Let
be the probability density of and assume
that is defined from to .
• Then the probability of having a value of less
than
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
5
Univariate Probability Distributions
• Therefore, we have
• We will also have
• The probability of obtaining a value in an interval
between and where
is
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
6
Univariate Probability Distributions
• The expected value or mean of
is defined as
• Further we can manipulate expectations by
Where
and
are constants.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
7
Univariate Probability Distributions
• Variance is a measure of the expected variation of
variable around its mean and is defined as,
• It can also be written as
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
8
Univariate Probability Distributions
• We can further write
• From this we can construct a new r.v.
,
and if the mean and variance of are zero and one
correspondingly then we have,
• This proves very useful in constructing random
variables with desired mean and variance.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
9
Univariate Probability Distributions
• Mean and variance are the first two central
moments. Third and fourth central moments, also
known as skewness and kurtosis are defined by,
• Looking closely at the formulas we will see that,
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
10
Univariate Probability Distributions
• As an example we can consider the normal
distribution with parameters, and .
• It is defined by
• The normal distribution moments are:
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
11
Bivariate Distribution
• When considering two random variables
we can define the bivariate density
and
• Covariance, the most commonly used measure of
dependence between two random variables is
defined as,
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
12
Bivariate Distribution
• Covariance has the following properties,
• We can define correlations as,
• We have
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
13
Bivariate Distribution
• A perfect positive linear relationship between
and exists if
• A perfect positive linear relationship between
and exists if
• The correlation is always bounded between -1
and +1.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
14
Conditional Distribution
• If we want to describe an RV y using information
on another RV x we can use conditional
distribution of y given x
• This definition can be used to define conditional
mean and variance
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
15
Conditional Distribution
• If x and y are independent then
• This means
.
• The conditional moments can be rewritten as
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
16
Sample Moments
• Here we want to introduce the standard methods for
estimating the moments introduced earlier.
• Consider sample of T observations of the variable x.
• We can estimate the mean using the sample average,
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
17
Sample Moments
• Similarly, we can estimate the variance using,
• Sometimes, the sample variance formula uses
instead of
, however, unless is very small
the difference is negligible.
• Skewness and kurtosis can be estimates as,
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
18
Sample Moments
• The sample covariance between two random
variables can be estimated by,
• The sample correlation between two random
variables can be found in a similar fashion by,
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
19
The Linear Model
• Linear models of the type below is often used by
risk managers,
• Where
and and are assumed to be
independent or sometimes uncorrelated.
• If we know the values of then we can use the
linear model to predict ,
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
20
The Linear Model
• This gives us,
• We also have that,
• This means that,
• In the linear model the variances are linked by
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
21
The Linear Model
• Consider observation in the linear model.
• If we have a sample of
estimate
observation we can
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
22
The Linear Model
• In the morel general linear model with different
variables we have,
• Minimizing the sum of squared errors,
provides the ordinary least squared (OLS) estimate
of ,
• OLS is built in to most common software
packages. In Excel OLS is done using LINEST
function.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
23
The Importance of Data Plots
• Linear relationship between two variables can be
deceiving
• Consider the four (artificial) data sets in table below
which are known as Anscombe’s quartet
• All four data sets have 11 observations
• Observations in the four data sets are clearly
different from each other
• The mean and variance of the x and y variables is
exactly the same across the four data sets
• The correlation between x and y are also the same
across the four pairs of variables
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
24
The Importance of Data Plots
• We also get the same regression parameter estimates in
all the four cases
• Figure 3.1 scatter plots y against x in the four data sets
with the regression line included. We see,
• A genuine linear relationship as in the top-left panel
• A genuine nonlinear relationship as in the top-right panel
• A biased estimate of the slope driven by an outlier
observation as in the bottom-left panel
• A trivial relationship, which appears as a linear
relationship again due to an outlier
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
25
Table 3.1: Anscombe's Quartet
I
II
III
IV
x
y
x
y
x
y
x
y
10
8.04
10
9.14
10
7.46
8
6.58
8
6.95
8
8.14
8
6.77
8
5.76
13
7.58
13
8.74
13
12.74
8
7.71
9
8.81
9
8.77
9
7.11
8
8.84
11
8.33
11
9.26
11
7.81
8
8.47
14
9.96
14
8.1
14
8.84
8
7.04
6
7.24
6
6.13
6
6.08
8
5.25
4
4.26
4
3.1
4
5.39
19
12.5
12
10.84
12
9.13
12
8.15
8
5.56
7
4.82
7
7.26
7
6.42
8
7.91
5
5.68
5
4.74
5
5.73
8
6.89
Mean
9.0
7.5
9.0
7.5
9.0
7.5
9.0
7.5
Variance
11.0
4.1
11.0
4.1
11.0
4.1
11.0
4.1
Moments
Correlation
0.82
0.82
0.82
0.82
a
3.00
3.00
3.00
3.00
b
0.50
0.50
0.50
0.50
Regression
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Figure 3.1
Scatter Plot of Anscombes Four Data Sets with
Regression Lines
I
14
10
10
8
8
y
12
y
12
6
6
4
4
2
2
0
0
5
10
x
15
0
20
III
14
0
5
10
10
8
8
y
12
6
6
4
4
2
2
0
10
x
15
20
15
20
IV
14
12
y
II
14
0
0
5
10
x
15
20
0
5
10
x
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
26
27
Univariate Time Series Models
• It studies the behavior of a single random variable
observed over time
• These models forecast the future values of a
variable using past and current observations on the
same variable
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
28
Autocorrelation
• Autocorrelation measures the dependence between
the current value of a time series variable and the
past value of the same variable.
• The autocorrelation for lag is defined as
• It captures the linear relationship between today’s
value and the value days ago
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
29
Autocorrelation
• Assuming
represents the series of
returns, the sample autocorrelation measures the
linear dependence between today’s return, , and
the return days ago,
• To see the dynamics of a time series it is very
useful to plot the autocorrelation function which
plot on the vertical axis against on the
horizontal axis.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
30
Autocorrelation
• The statistical significance of a set of
autocorrelations can be formally tested using the
Ljung-Box statistic.
• It tests the null hypothesis that the autocorrelation
for lags 1 through m are all jointly zero via
• Where
denotes the chi-squared distribution.
CHIINV(.,.) can be used in Excel to find the
critical values.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
31
Autoregressive (AR) Models
• If a pattern is found in the autocorrelations then
we want to match that pattern in our forecasting
model.
• The simplest model for this purpose is the
autoregressive model of order 1, which is defined
as
• Where,
,
, and
assumed to be independent for all
and
.
are
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
32
Autoregressive (AR) Models
• The condition mean forecast for one period ahead
under this models is,
• By using the AR formula repeatedly we can write,
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
33
Autoregressive (AR) Models
• The multistep forecast in the AR(1) model is
therefore given by
• If
by,
implies
then the (unconditional) mean is given
, which in the AR(1) model
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
34
Autoregressive (AR) Models
• When
,
.
• The (unconditional) variance is similarly,
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
35
Autoregressive (AR) Models
• To derive the ACF for AR(1) model without loss
of generality we can assume that
• Then we would get,
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
36
Figure 3.2
Autocorrelation Functions for AR(1) Models with
Positive
1.2
Ø=1
Ø = 0.99
Ø = 0.5
Ø = 0.1
1
0.8
0.6
0.4
0.2
0
10
20
30
40
50
60
70
80
90
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
100
37
Autoregressive (AR) Models
• Figure 3.2 shows examples of the ACF in AR(1)
models
• When <1 then the ACF decays to zero
exponentially
• The decay is much slower when = 0.99 than
when it is 0.5 or 0.1
• When
=1 then the ACF is flat at 1. This is the
case of a random walk
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Figure 3.3
Autocorrelation Functions for AR(1) Models with
Positive =-0.9
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
10
20
30
40
50
60
70
80
90
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
100
38
39
Autoregressive (AR) Models
• Figure 3.3 shows the ACF of an AR(1) when
. =-0.9
• When <0 then the ACF oscillates around zero
but it still decays to zero as the lag order increases
• The ACFs in Figure 3.2 are much more common
in financial risk management than are the ACFs in
Figure 3.3
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
40
Autoregressive (AR) Models
• The simplest extension to the AR(1) model is the
AR(2) model defined as,
• The ACF of the AR(2) is
• Because for example,
• So that,
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
41
Autoregressive (AR) Models
• In order to derive the first lag autocorrelation note
that the ACF is symmetric around
meaning
that,
• We therefore get that
• Which in turn implies that,
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
42
Autoregressive (AR) Models
• The general AR(p) model is simply defined as
• The
day ahead forecast can be built using
• Which is called the chain rule of forecasting.
• Note that when
then,
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
43
Autoregressive (AR) Models
• The partial autocorrelation function (PACF) gives
the marginal contribution of an additional lagged
term in AR models of increasing order.
• First estimate a series of AR models of increasing
order:
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
44
Autoregressive (AR) Models
• The PACF is now defined as the collection of the
largest order coefficients,
• Which can be plotted against the lag order just as
we did for the ACF.
• The optimal lag order p in the AR(p) can be
chosen as the largest p such that
is significant
in the PACF.
• Note that in the AR models the ACF decays
exponentially whereas the PACF drops abruptly.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
45
Moving Average Models
• In AR models the ACF dies off exponentially but
in finance there are cases such as bid-ask spreads
where the ACFs die off abruptly.
• These require a different type of model.
• We can consider MA(1) model defined as
• Where and
and
.
• Note that
»
are independent of each other
and
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
46
Moving Average Models
• To derive the ACF of the MA(1) assume without
loss of generality that
we then have,
• Using the variance expression from before, we get
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
47
Moving Average Models
• The MA(1) model must be estimated by numerical
optimization of the likelihood function.
– First set the unobserved
– Second, set parameter starting values for , ,
and .
– We can use the average of for , use 0 for
and use the sample variance of for
• Now compute time series of residuals via
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
48
Moving Average Models
• If we assume that
is normally distributed then
• Since are assumed to be independent over time
we have,
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
49
Moving Average Models
• We can use an iterative search (using for example
Solver of Excel) to find the parameters’ (
)
estimates for the MA(1)
• Once the parameters are estimated we can use the
model for forecasting. The conditional mean
forecast is,
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
50
Moving Average Models
• The general MA(q) model is defined,
• The ACF for MA(q) is non-zero for the first q lags
and then drops abruptly to zero.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
51
ARMA Models
• We can combine AR and MA models.
• ARMA models often enables us to forecast in a
parsimonious manner.
• ARMA(1,1) is defined as
• The mean of the ARMA(1,1) times series is
• When
,
• Rt will tend to fluctuate around the mean
• Rt is mean-reverting in this case
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
52
ARMA Models
• Using the fact that
, variance is
• Which implies that,
• The first order autocorrelation is given from
• In which we assume again that
.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
53
ARMA Models
• We can write,
• So that,
• For higher order autocorrelation the MA term has
no effect and we get the same structure as in the
AR(1),
• The general ARMA(p,q) model is,
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Random Walks, Unit Roots, and
ARIMA
• Let , be the closing price of an asset and let
,
so that the log returns are defined by
• The random walk (or martingale) model is now
defined as
• By iteratively substituting in lagged log prices we
can write,
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
54
Random Walks, Unit Roots, and
ARIMA
• In Random Walk model the conditional mean and
variance are given by,
• Equity returns typically have a small positive
mean corresponding to a small positive drift in the
log price. This motivates RW with drift:
• Substituting in lagged prices back to time 0,
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
55
Random Walks, Unit Roots, and
ARIMA
•
follows an ARIMA(p,1,q) model if the first
difference,
, follows a mean reverting
ARMA(p,q) model.
• In this case we say that has a unit root.
• The random walk model has a unit root as well
because
which is a ARMA(0,0) model
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
56
57
Pitfall #1: Spurious Mean-Reversion
• Consider the AR(1) model
• Note, when
, AR(1) model has a unit root
and becomes the random walk model
• The OLS estimator contains a small sample bias
in dynamic models
• In an AR(1) model when the true coefficient is
close or equal to 1, the finite sample OLS
estimate will be biased downward.
• This is known as the Hurwitz bias or the DickeyFuller bias
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
58
Pitfall #1: Spurious Mean-Reversion
• Econometricians are skeptical about technical trading
analysis as it attempts to find dynamic patterns in
prices and not returns
• Asset prices are likely to have a very close to 1
• But it is likely to be estimated to be lower than 1,
which in turn suggests predictability
• Asset returns have a
close to zero and its estimate
does not suffer from bias
• Dynamic patterns in asset returns is much less likely to
produce false evidence of predictability than is
dynamic patterns in asset prices
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
59
Testing for Unit Roots
• Asset prices often have a
very close to 1
• We need to determine whether = 0.99 or 1 because
the two values have very different implications for
long term forecasting
• = 0.99 implies that the asset price is predictable
whereas = 1 implies it is not
• Consider the AR(1) model with and without a constant
term
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
60
Testing for Unit Roots
• Unit root tests have been developed to assess the
null hypothesis
• against the alternative hypothesis that
• When the null hypothesis H0 is true, so that =1,
the unit root test does not have the usual normal
distribution even when T is large
• OLS estimation of to test =1 using the usual ttest, likely leads to rejection of the null hypothesis
much more often than it should
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
61
Multivariate Time Series Models
• Multivariate time series analysis consider risk
models with multiple related risk factors or models
with many assets
• This section will introduce the following topics:
 Time series regressions
 Spurious relationships
 Cointegration
 Cross correlations
 Vector autoregressions
 Spurious causality
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
62
Time Series Regression
• The relationship between two time series can be
assessed using the regression analysis
• But the regression errors must be scrutinized
carefully
• Consider a simple bivariate regression of two highly
persistent series
• Example: the spot and futures price of an asset
• To diagnose a time series regression model, we need
to plot the ACF of the regression errors, et.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
63
Time Series Regression
• If ACF dies off only very slowly, then we need to firstdifference each series and run the regression
• Now use the ACF on the residuals of the new
regression and check for ACF dynamics
• The AR, MA, or ARMA models can be used to model
any dynamics in et.
• After modeling and estimating the parameters in the
residual time series, et, the entire regression model
including a and b can be reestimated using MLE.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
64
Pitfall #2: Spurious Regression
• Consider two completely unrelated times series—
each with a unit root
• They are likely to appear related in a regression that
has a significant b coefficient
• Let s1t and s2t be two independent random walks
• where
are independent of each other and
independent over time.
• True value of b is zero in the time series regression
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
65
Pitfall #2: Spurious Regression
• However standard t-tests will tend to conclude that b is
nonzero when in truth it is zero.
• This problem is known as spurious regression
• So, use ACF to detect spurious regression
• If the relationship between s1t and s2t is spurious then
the error term, et; will have a highly persistent ACF and
the regression in first differences will not show a
significant estimate of b
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
66
Cointegration
• Relationships between variables with unit roots are not
always spurious.
• A variable with a unit root is also called integrated
• If two variables that are both integrated have a linear
combination with no unit root then we say they are
cointegrated.
• Examples: long-run consumption and production in an
economy
• The spot and the futures price of an asset that are
related via a no-arbitrage condition.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Cointegration
• The pairs trading strategy consists of two stocks
whose prices tend to move together.
• If prices diverge then we buy the temporarily
cheap stock and short sell the temporarily
expensive stock and wait for the typical
relationship between the prices to return
• Such a strategy hinges on the stock prices being
cointegrated
• Consider a simple bivariate model where
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
67
68
Cointegration
• Note that s1t has a unit root and that the level of s1t
and s2t are related via b.
• Assume that
are independent of each
other and independent over time.
• The cointegration model can be used to preserve
the relationship between the variables in the longterm forecasts
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
69
Cross-Correlations
• Consider two financial time series, R1,t and R2,t
• They can be dependent in three possible ways:
•
can lead
(e.g.,
)
•
can lag
(e.g.,
),
• They can be contemporaneously related
(e.g.,
)
• We use cross-correlation matrices to detect all these
possible dynamic relationships
• The sample cross-correlation matrices are the
multivariate analogues of the ACF function
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
70
Cross-Correlations
• For a bivariate time series, the cross-covariance
matrix for lag is
• The two diagonal terms are the autocovariance
function of R1,t, and R2,t, respectively
• In the general case of a k-dimensional time series, we
have
• where Rt is now a k by 1 vector of variables
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
71
Cross-Correlations
• Detecting lead and lag effects is important
• For example, when relating an illiquid stock to a liquid
market factor.
• The illiquidity of the stock implies price observations that
are often stale, which in turn will have a spuriously low
correlation with the liquid market factor.
• The stale equity price will be correlated with the lagged
market factor and this lagged relationship is used to
compute a liquidity-corrected measure of the dependence
between the stock and the market
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
72
Vector Autoregressions (VAR)
• Consider a first-order Vector Autoregression, call it
VAR(1)
• where Rt is again a k by 1 vector of variables
• The bivariate case is simply
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
73
Vector Autoregressions (VAR)
• In the VAR, R1,t and R2,t are related via their
covariance
• The VAR only depends on lagged variables so, it
is immediately useful in forecasting.
• If the variables included on the right-hand-side of
each equation in the VAR are the same then the
VAR is called unrestricted
• If so, OLS can be used equation-by-equation to
estimate the parameters.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
74
Pitfall #3: Spurious Causality
• We may want to see if the lagged value of
,
namely
, is causal for the current value of
• If so,
can be used in forecasting
• Consider a simple regression of the form
• This regression may easily lead to false
conclusions if
is persistent and so depends
on its own past value
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
75
Pitfall #3: Spurious Causality
• To truly assess if
causes
we need to
check if past
was useful for forecasting current
.
once the past
has been accounted for
• This question can be answered by running a VAR
model:
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
76
Pitfall #3: Spurious Causality
• Now we can define Granger causality as follows:
• In some cases several lags of
may be needed on the
right-hand side of the equation for
• We may need more lags of
in the equation for
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
77
Summary
• Financial asset prices and portfolio values can be
viewed as examples of very persistent time series
• The three most important issues are
• Spurious detection of mean reversion-erroneously
finding that a variable is mean-reverting when it is
truly a random walk
• Spurious regression-erroneously finding that a variable
x is significant when regressing y on x
• Spurious detection of causality-erroneously finding that
the current value of x causes future values of y when in
reality it cannot
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen